■68 The Ohio Journal of Science [Vol. XVI, No. 2, 



Obviously no algebaric solution can be obtained from 

 equations (3) and (4) for k and X in terms of the computed values 

 j8 and e, and hence a resort to tables is necessary. The values of 

 j8 and e for values of k from to 0.0335 and of X from -0.040 to 

 +0.100 have been computed.* The process of determining the 

 constants of the translated normal curve consists first in com- 

 puting /5 and e from the given data, and then in entering the 

 table and interpolating for the corresponding values of k and 

 X.f On substituting these values in (2) the value of a can be 

 found and thence on multiplying a by k the position of the 

 median of the distribution is obtained. 



The sign of k is determined by the sign of the third moment 

 about the mean ixz, that is, by the direction of the skewness or 

 asymetry. For positive skewness the mean must lie to the right 

 of the median and hence ixi , the first moment about the mean, 

 must be positive which necessitates a positive sign for k. 

 Therefore, the sign of k is the same as that of the skewness. 



To fit a curve to the given data, after the constants have 

 been determined it is necessary to find, by solving a cubic 

 equation for each value, the values of t corresponding to the 

 x's of the respective classes. The cubic is 



aXt^ + a/ct" + at — x = 



Any of the various methods of approximating to the solution 

 of a cubic may be used in solving these equations. 



The area of each class can now be obtained by computing 

 the corresponding areas under the standard normal curve from 

 a table of the probability integral. 



The Method of Interpolation. The actual fitting of the curve 

 can now be readily accomplished. J The distinctively geomet- 

 rical operation is the interpolation for the values of X and k for 

 a given pair of values of jS and e. 



Within the limits of the table § the curves resulting from the 

 assignment of a constant value to jS are practically straight 



*Only a part of the original table appears in the accompanying talile. The 

 original values were computed to four places of decimals, but three jilace numbers 

 are sufficient to illustrate the method of approximating to the solution. 



fCompare "Tables for Statisticians and Biometricians," Caml)ridge Uni- 

 versity Press, 1914. 



JFor the statistical details see Elderton, 1. c. 

 §As may be seen on examining the Table. 



